Efficiency versus robustness: The case for minimum Hellinger distance and related methods. (English) Zbl 0807.62030

Summary: It is shown how and why the influence curve poorly measures the robustness properties of minimum Hellinger distance estimation. Rather, for this and related forms of estimation, there is another function, the residual adjustment function, that carries the relevant information about the trade-off between efficiency and robustness. It is demonstrated that this function determines various second-order measures of efficiency and robustness through a scalar measure called the estimation curvature. The function is also shown to determine the breakdown properties of the estimators through its tail behavior.
A 50% breakdown result is given. It is shown how to create flexible classes of estimation methods in the spirit of \(M\)-estimation, but with first-order efficiency (or even second-order efficiency) at the chosen model, 50% breakdown and a minimum distance interpretation.


62F35 Robustness and adaptive procedures (parametric inference)
62F12 Asymptotic properties of parametric estimators
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